Problem: $f(x) = 25x^2-10x+1 $ What is the value of the discriminant of $f$ ?
Solution: The ${\text{discriminant}}$ is a part of the quadratic formula. The sign of the discriminant tells us whether there are two roots, one root, or no roots. $\dfrac{-b\pm{\sqrt{\overbrace{{b^2-4ac}}^{\text{discriminant}}}}}{2a}$ Discriminant Roots Positive Two real roots Zero One repeated real root Negative No real root Let's find the discriminant of $f$ is given by: $\begin{aligned} {b^2-4ac}&=(-10)^2-4\cdot25\cdot1 \\\\ &=100-100 \\\\ &={0} \end{aligned}$ So how many $x$ -intercepts does the graph of $f$ have? Since the discriminant is ${0}$, $f$ has $1$ $x$ -intercept. In conclusion: The discriminant of $f$ is ${0}$. The graph of $f$ has $1$ $x$ -intercept.